Stability Analysis of Numerical Methods for a 1.5-Layer Shallow-Water Ocean Model

Guang-An Zou,Mu Mu,Bo Wang

doi:10.1155/2013/478054

Abstract

A 1.5-layer reduced-gravity shallow-water ocean model in spherical coordinates is described and discretized in a staggered grid (standard Arakawa C-grid) with the forward-time central-space (FTCS) method and the Leap-frog finite difference scheme. The discrete Fourier analysis method combined with the Gershgorin circle theorem is used to study the stability of these two finite difference numerical models. A series of necessary conditions of selection criteria for the time-space step sizes and model parameters are obtained. It is showed that these stability conditions are more accurate than the Courant-Friedrichs-Lewy (CFL) condition and other two criterions (Blumberg and Mellor, 1987; Casulli, 1990, 1992). Numerical experiments are proposed to test our stability results, and numerical model that is designed is also used to simulate the ocean current.

Highlights

The shallow-water model is a set of partial differential equations (PDEs), which derived from the principles of conservation of mass and conservation of momentum
Because the horizontal length scale is much greater than the vertical length scale, under this condition, the conservation of mass implies that the vertical velocity of the fluid is very small
The situations in fluid dynamics where the horizontal length scale is much greater than the vertical length scale are very common; that is to say, the vertical acceleration of the fluid can be negligible

Summary

Introduction

The shallow-water model is a set of partial differential equations (PDEs), which derived from the principles of conservation of mass and conservation of momentum (the NavierStokes equations). Because the horizontal length scale is much greater than the vertical length scale, under this condition, the conservation of mass implies that the vertical velocity of the fluid is very small. It can be shown from the momentum equations that horizontal pressure gradients are due to the displacement of the pressure surface (or free surface) in a fluid, and that vertical pressure gradients are nearly hydrostatic [1]. We use the discrete Fourier analysis method and the Gerschgorin circle theorem to study the stability of the shallow-water numerical models and give a series of necessary conditions for the selection criteria of time step size.

Matematical Model

Finite Difference Schemes

Stability Analysis

Numerical Experiments

Conclusions